Maximal subgroups of GLn(D)

Maximal Subgroups of Subnormal Subgroups ofGLn(D) with Finite Conjugacy Classes

“…If M is a maximal subgroup of GL n D , then it is possible that a ∈ M\Z M has finitely many conjugates in M. Indeed, if F M satisfies a polynomial identity (specially when D F < ), where F = Z D , then surely there exists an element in M\Z M which has finitely many conjugates by a theorem of Passman [14, p. 189] and Theorem 1. In [1,13] it was proved that * ∪ * j is a non-abelian solvable maximal subgroup of the real quaternions division ring; clearly, this group has a noncentral element with finitely many conjugates.…”

Nilpotent maximal subgroups of GLn(D)

“…The case n = 1 was done in Corollary 5, so we can assume that n 2. Observing Theorems 12 and 13 of [1] we can assume that M is not absolutely irreducible and D is infinite dimensional over its center F . First we show that M is irreducible.…”

“…In this paper we investigate some properties of maximal subgroups of the general skew linear group. The structure of such groups have been studied in various papers (e.g., see [1][2][3][4]). An interesting question which has not been answered yet is whether the multiplicative group of every noncommutative division ring has a maximal subgroup.…”

“…In [4] it is conjectured that the multiplicative group of a division ring contains no nilpotent maximal subgroup. In connection with this conjecture, in [1] it is proved that if D is a division ring with center F and M is a nilpotent maximal subgroup of D * such that F [M]\F contains an algebraic element over F , then M is abelian. Here using crossed products we omit the above condition and prove a more general statement, namely for any natural number n and any infinite division ring D, the group GL n (D) contains no nonabelian nilpotent maximal subgroup.…”

SOLUBLE MAXIMAL SUBGROUPS IN GLn(D)

“…The structure of nilpotent maximal subgroups of GL n (D) is studied in [3], and it is shown that the nilpotent maximal subgroups of GL n (D) are abelian. To mention further results in this direction, the authors in [1] give a classification of all reducible maximal subgroups of GL n (D), and conjecture that GL n (D) contains no soluble maximal subgroups if n > 1. Here, as a consequence of our main result, we shall see that this conjecture is true for non-abelian soluble maximal subgroups.…”